Rép. : Bernoulli numbers

[Mohammed BOUAYOUN]

if b[n_] := Numerator[BernoulliB[2n]/(2n(2n+1))]    

For[n=1, n <= 1400, n++, 
If[ a[n] != b[n], Print[n, " ", a[n]/b[n]] ]
]

produces 

x(1)  =610   y(1)=  37
x(2) =1276  y(2)=37
x(3) =1287  y(3)=103
x(4) =1327  y(4)=59

and if b[n_] := Numerator[BernoulliB[2n]/(2n(2n-1))] 
produces

z(1)=547       w(1)=37
z(2)=1185     w(2)=103
z(3)=1240     w(3)=37
z(4)=   1269  w(4)=59

I remarque that

z(1)/w(1) + x(1)/y(1) = 32 (integer)
z(2)/w(2) + x(2)/y(2) = 45.9913409
z(3)/w(3) + x(3)/y(3) = 46,0086591 
z(4)/w(4) + x(4)/y(4) = 44 (integer)
z(5)/w(5) + x(5)/y(5) = 22 (integer)


I remarque that  z(2)/w(2) + x(2)/y(2)   +  z(3)/w(3) + x(3)/y(3)  = 45.9913409 + 46,0086591 = 92

I define sequence x(n) and y(n) and the sequence z(n)/w(n) + x(n)/y(n) if integer and another sequence z(n)/w(n) + x(n)/y(n)  is not integer

can anyone extend the sequence 

b[n_] := Numerator[BernoulliB[2n]/(2n(2n+1))]    
a[n_] := Numerator[BernoulliB[2n]/(2n)]          

For[n=1, n <= ?????????, n++, 
If[ a[n] != b[n], Print[n, " ", a[n]/b[n]] ]
]

Thank for your help

Merci beaucoup de votre aide

>>> "N. J. A. Sloane" <njas at research.att.com> 02/03 7:56  >>>

Dear SeqFans,   Michael Somos made an interesting discovery.
A001067 and A046968 are not the same!
In fact: (speaking baby Mathematica talk, which is
all that i know): 

a[n_] := Numerator[BernoulliB[2n]/(2n)]          (* A001067 *)
b[n_] := Numerator[BernoulliB[2n]/(2n(2n-1))]    (* A046968 *)

For[n=1, n <= 580, n++, 
If[ a[n] != b[n], Print[n, " ", a[n]/b[n]] ]
]

produces one line of output

                  574 37

In other words, the sequence of values of n such that
A001067(n) differs from A046968(n) starts
574, ...
and the associated ratios begin
37, ...

Could someone with more computing power than I have extend
these two sequences?  PARI gives the same result.  Maple dies.


Thanks

Neil


Neil J. A. Sloane
AT&T Shannon Labs, Room C233, 
180 Park Avenue, Florham Park, NJ 07932-0971
Email: njas at research.att.com
Office: 973 360 8415; fax: 973 360 8178
Home page: http://www.research.att.com/~njas/
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